Introduction to the DFT and Frequency Sampling.

Cuthbert Nyack
The Frequency sampling approach to designing FIR filters is conceptually the simplest approach to FIR filters. Unfortunately it is also the most tedious to apply. The images below and the applets on the following pages attempt to show why this is the case.

In principle, the proceedure is as follows.
First determine the frequency response of the desired filter LP, HP etc.
Next sample this frequency response with the number of samples being equal to the length of the desired FIR filter.
Now use the IDFT of the frequency samples to find the Impulse response of the filter.
Using the impulse response and the Z transform, plot the DTFT frequency response of the impulse response and verify that the resulting frequency response is close enough to the desired frequency response. If not then adjustments will need to be made to the desired frequency response using transition samples or to the impulse response using windows in order to get an acceptable frequency response.

The Impulse response of an Ideal low pass filter is:-
That of an ideal high pass filter is:-
and that of an ideal band pass filter is:-

The next 2 images attempt to show why this method can become very tedious. In the first image one sees that the frequency response calculated from the impulse response shown in green passing smoothly through the samples shown by the red x's. This happens provided the desired frequency response does not have major discontinuities in its value or its derivatives.
When one tries to increase the attenuation in the stop band, then the image below shows what can happen. The frequency response still passes through the samples but now one or more large peaks may appear between the samples. After experimenting for a while, one begins to feel like someone who is trying to stay afloat by using a basket as a flotation device. Good frequency responses can only be obtained by numerical methods. However the fact that almost every numerical method which can be, have been applied to this problem, suggest the tediousness involved.
In the following pages applets are included which one can use to interactively adjust the frequency sample values, in order to get a feeling for what sort of frequency samples reproduce an acceptable frequency response. Windows and Transition samples are also illustrated.
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COPYRIGHT 2008 Cuthbert Nyack.